2-cocycle for a Lie ring action

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Definition

Suppose L is a Lie ring, M is an abelian group, and \varphi:L \to \operatorname{End}(M) is a Lie ring homomorphism from L to the ring of endomorphisms of M, where the latter gets the usual Lie ring structure from its structure as an associative ring.

Explicit definition

A 2-cocycle for this action is a homomorphism of groups f:L \times L \to M (where L \times L is the external direct product) satisfying the additional condition:

\! \varphi(x)(f(y,z)) - \varphi(y)(f(x,z)) + \varphi(z)(f(x,y)) - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L

If we suppress \varphi and denote the action by \cdot, this can be written as:

\! x \cdot f(y,z) - y \cdot f(x,z) + z \cdot f(x,y) - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L

Definition in terms of the general definition of cocycle

A 2-cocycle for a Lie ring action is a special case of a cocycle for a Lie ring action, namely n = 2.

Group structure

The set of 2-cocycles for the action of L on M forms a group under pointwise addition.

Particular cases and variations

Case or variation Condition for an additive homomorphism to be a 2-cocycle Further information
M = L and \varphi is the adjoint action, i.e., \varphi(x) is the inner derivation induced by x \! [x,f(y,z)] - [y,f(x,z)] - [z,f(x,y)] - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L
\varphi is the zero map \! f(x,[y,z]) = f([x,y],z) + f(y,[x,z]) \ \forall \ x,y,z \in L 2-cocycle for trivial Lie ring action
L is an abelian Lie ring \! y \cdot f(x,z) = x \cdot f(y,z) + z \cdot f(x,y) \ \forall \ x,y,z \in L
\varphi is the zero map and L is an abelian Lie ring No additional condition

Related notions